Question

In the figure, ▢ABCD is a cyclic quadrilateral. If m(arc ABC) = 230°, then find ∠ABC, ∠CDA, ∠CBE.

Answer 1

m(arc ABC) = 230°   .....(i) [Given]

∴ m(arc ADC) + m(arc ABC) = 360°   .......[Degree measure of a circle is 360°]

∴ m(arc ADC) = 360° – m(arc ABC)

∴ m(arc ADC) = 360° – 230°   .......[From (i)]

∴ m(arc ADC) = 130°

∠ABC = `1/2` m (arc ADC)   ......[Inscribed angle theorem]

= `1/2 xx 130^circ`

= 65°

Now, ∠CDA = `1/2` m (arc ABC)   ......[Inscribed angle theorem]

∴ ∠CDA = `1/2 xx 230^circ`

∴ ∠CDA = 115°    ......(ii)

∠CBE = ∠CDA    ......(iiii) [The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle]

∴ ∠CBE = 115°     .....[From (ii) and (iii)]

∴ ∠ABC = 65°, ∠CDA = 115°, ∠CBE = 115°.